Preheat oven to 350 ºF. Butter a 15-by-10-inch jelly-roll pan; line with parchment paper, leaving a 2-inch overhang on the two shorter sides.

In a medium bowl, whisk together butter and sugar until combined; whisk in egg, vanilla, and salt until combined. Add flour and cocoa, and mix just until smooth. Spread in prepared baking pan. Bake until cake is dry to the touch and edges begin to pull away from the sides, 10 to 12 minutes. Cool completely in pan.

Using paper overhang, lift cake onto a work surface. With a serrated knife, halve crosswise. Place one half of cake, flat side down, on a large piece of plastic wrap. Spread with ice cream, then top with remaining half of cake, flat side up. Return ice-cream sandwich to baking pan, and wrap tightly in plastic. Freeze until firm, about 2 hours.

Unwrap sandwich; using a serrated knife, cut into 8 rectangles, wiping off blade between each cut. Serve ice-cream sandwiches immediately.